Logistic‐growth models measuring density feedback are sensitive to population declines, but not fluctuating carrying capacity

Abstract Analysis of long‐term trends in abundance of animal populations provides insights into population dynamics. Population growth rates are the emergent interplay of inter alia fertility, survival, and dispersal. However, the density feedbacks operating on some vital rates (“component feedback”) can be decoupled from density feedbacks on population growth rates estimated using abundance time series (“ensemble feedback”). Many of the mechanisms responsible for this decoupling are poorly understood, thereby questioning the validity of using logistic‐growth models versus vital rates to infer long‐term population trends. To examine which conditions lead to decoupling, we simulated age‐structured populations of long‐lived vertebrates experiencing component density feedbacks on survival. We then quantified how imposed stochasticity in survival rates, density‐independent mortality (catastrophes, harvest‐like removal of individuals) and variation in carrying capacity modified the ensemble feedback in abundance time series simulated from age‐structured populations. The statistical detection of ensemble density feedback from census data was largely unaffected by density‐independent processes. Long‐term population decline caused from density‐independent mortality was the main mechanism decoupling the strength of component versus ensemble density feedbacks. Our study supports the use of simple logistic‐growth models to capture long‐term population trends, mediated by changes in population abundance, when survival rates are stochastic, carrying capacity fluctuates, and populations experience moderate catastrophic mortality over time.


| INTRODUC TI ON
Compensatory density feedback describes a population's ability to return to the environment's carrying capacity in response to an increase in population size (sensu Herrando-Pérez et al., 2012b)-see the full glossary of terms in Table 1. Density feedback is a phenomenon driven by adjustments to individual fitness imposed by variation in per-capita resource availability, dispersal, and associated trophic and social interactions, including competition, predation, and parasitism (Eberhardt et al., 2008;Herrando-Pérez et al., 2012a;Matthysen, 2005). As survival and fertility rates ebb and flow over time, it is theoretically possible to detect the presence, and quantify the strength, of density feedbacks in population growth rates using abundance time series. Such "census data" results from populations monitored at semiregular intervals over a sufficient period relative to the generation length of the species under investigation Herrando-Pérez et al., 2012a). There is now considerable evidence that survival and reproduction track population trends in many vertebrate (Eberhardt, 2002;Morrison et al., 2021;Owen-Smith & Mason, 2005;Paradis et al., 2002;Pardo et al., 2017) and invertebrate (Bonsall & Benmayor, 2005;Ma, 2021;Marini et al., 2016;McGeoch & Price, 2005) species. Therefore, given the irreplaceable importance of long-term monitoring of population size in applied ecology and conservation (Bonebrake et al., 2010;Di Fonzo et al., 2016;Herrando-Pérez et al., 2012a;Micheli et al., 2020), assessing the strength of compensatory signals in censuses of population abundance remains an essential tool in the ecologist's toolbox (Hostetler & Chandler, 2015;Johnson et al., 2022;Ponciano et al., 2018;Rueda-Cediel et al., 2015;Thibaut & Connolly, 2020).
The family of self-limiting population-growth models including logistic growth curves ("phenomenological models" hereafter) (Eberhardt et al., 2008) use census data to quantify the net effect of population size N on the per capita rate of exponential population change r (Berryman & Turchin, 2001). Expressed as a proportional change in N between two time (t) steps (e.g., years or generations), the assumption is that the discrete-time metric r t = log e (N t + 1 /N t ) summarizes the combination or "ensemble" (Herrando-Pérez et al., 2012a) of all "component" density feedbacks operating on survival, fertility, and dispersal (Münster-Swendsen & Berryman, 2005). The problem is that population growth rates can be insensitive to variation in particular demographic rates (Battaile & Trites, 2013;Bürgi et al., 2015;Kolb et al., 2010). Thus, across 109 observed censuses of bird and mammal populations, the strength of "component density feedback" (on demographic rates) explained only <10% of the strength of "ensemble density feedback" (on population growth rate) using logistic models after controlling for time-series length and species' body size (Herrando-Pérez et al., 2012a). The potential reasons for such decoupling include observation error (Abadi et al., 2012;Knape & de Valpine, 2012), fluctuating age structure (Hoy et al., 2020), unequal contribution to density feedbacks by age-structured individuals (Gamelon et al., 2016), shifting nonstationarity among vital rates (Layton-Matthews et al., 2019), immigration (Lieury et al., 2015), spatial heterogeneity (Thorson et al., 2015), and environmental state shifts (Turchin, 2003;Wu et al., 2007).
Determining the partial effects of different underlying mechanisms responsible for the decoupling of component and ensemble density feedbacks is virtually impossible for population censuses of real species. This limitation occurs for two main reasons: (1) the multiple, density-dependent and -independent mechanisms generating population fluctuations change themselves through time-a condition known as "nonstationarity" (sensu Turchin, 2003), and (2) the full set of those mechanisms is often unknown and/or not measured in wild populations. To build a fully controlled simulation environment that incorporated most mechanisms a priori determined to affect component-ensemble decoupling, we built stochastic, agestructured population models with known, component density feedbacks on survival. We imposed nonstationarity to population size via multiple demographic scenarios emulating density-independent mortality and temporal variation in carrying capacity. We then sim- Specifically, we simulated the dynamics of 21 long-lived vertebrates from a range of taxonomic/functional groups and body sizes, adjusting component feedbacks in each case to elicit initially stable dynamics. Theoretically, the strength of the ensemble signal (density feedback on population growth rate) must track the strength of the component signal (density feedback on survival), if survival has a demographic impact, mediated by population size, on the long-term population trends. To test our hypothesis, we then imposed nonstationarity to each time series in the form of two density-independent processes (catastrophic and harvest-like mortality; fluctuating carrying capacity) to assess the extent by which those processes mask the effect of the component signal on the ensemble signal.

| ME THODS
Our overarching aim was to simulate populations of long-lived species and their time series of abundance with known component feedback in survival (as well as in fertility in some cases to assess whether two component feedbacks altered our conclusions-they do not; see Appendix S1) and various sources of nonstationarity. Below, we describe the set of test species (Section 2.1), the simulation of the base population models for each species (Section 2.2) and of the component density feedbacks on survival within the base population models (Section 2.3), how we imposed (Section 2.4) and measured (Section 2.5) nonstationarity in the resultant time series of population abundance, how we simulated the abundance time series from the base population models (Section 2.6), the demographic scenarios TA B L E 1 Glossary of terms used in the paper. Italicized, boldface terms in the definition column indicate terms defined elsewhere in the table.

Carrying capacity
Maximum population density (commonly denoted K) a given environment can sustain indefinitely, so describing the equilibrium population density as determined by available resources. K can be exceeded, but only temporarily. Mathematically, K equates to the long-term mean population density where the per capita rate of exponential population change (r) approaches zero Berryman and Turchin (2001); Berryman (1999) Cohort Group of individuals that were born at the same time. Used to assign individuals to the same age class within a Leslie matrix including specific demographic rates Gotelli (2008) Compensation (compensatory) A density feedback whereby population density is negatively correlated with population growth, fertility, survival or dispersal. Population density declines when the carrying capacity is exceeded and vice versa Neave (1953) Component density feedback Density feedback compensating or depensating single demographic rates Herrando-Pérez et al. (2012b) Demographic rate A measurable aspect of individual fitness expressed as a probability or a rate over a defined time period, including survival (probability of an individual surviving from time t to t + 1), fertility (number of offspring per female produced per unit time), and dispersal (number of individuals leaving a defined population per unit time) Levin et al. (2008) Density feedback (~density dependence a ) When social and trophic interactions modify demographic rates and the resulting change in demographic rates alters population density, "feeding" back to modify the intensity of those interactions Berryman (1989); Berryman et al. (2002)

Density-feedback strength
The degree to which a demographic rate or population rate of change varies with increasing or decreasing population density. In the Ricker logistic model, this is measured as the slope of the compensatory (negative) relationship between the per capita rate of exponential population change and population density ; Doncaster (2008) Density independence Present or past population density not affecting per-capita population growth rate and/or demographic rates Herrando-Pérez et al. (2012b); Smith (1935) Depensation (depensatory) A density feedback whereby population density is positively correlated with population growth, fertility, survival, or dispersal. It typically occurs at low population density far from carrying capacity, often referred to as an "Allee effect" Courchamp et al. (1999); Neave (1953) Ensemble density feedback Compensation or depensation acting on a population's overall growth rate, representing the sum of all component density feedbacks Herrando-Pérez et al. (2012a); Münster-Swendsen and Berryman (2005) Gompertz logistic Linear (compensatory), discrete-time relationship between the per capita rate of exponential population change (r) and the natural logarithm of population density Doncaster (2008); Medawar (1940);Nelder (1961) Leslie matrix Also known as a population demographic matrix, it represents the probability of transitioning from one age class to the next (survival), and producing new individuals in the first age class (fertility)

Nonstationarity (nonstationary)
Occurs when density-dependent and -independent mechanisms generating fluctuations in population density themselves change through time Dennis and Taper (1994); Turchin (2003) Per capita rate of exponential population change Often denoted r, this is the rate of population change calculated as the natural logarithm of the ratio of population densities at time t + 1 to t, where r = log e (N t + 1 /N t ). When r = 0, the population is stable; when r < 0, it declines; when r > 1, the population is increasing

Phenomenological
Model describing the long-term dynamics of population density (cycles, stability, instability) resulting from demographic processes

Population density
Often denoted N, this is the number of individuals in a population per unit area; when the area under consideration does not shift through time, population size can replace density per se in dynamical models Berryman (1999)

Return time
Time required for a population to return to carrying capacity following a disturbance Berryman (1999) (Continues) we considered to examine which conditions led to a decoupling of component and ensemble feedbacks (Section 2.7), the logistic models we used to quantify ensemble density feedbacks from the projected time series of abundance (Section 2.8), and how we compared the strength of the component and ensemble density-feedback signals (Section 2.9). See Figure 1 for a detailed schematic of the process.

| Test species
Because the variability in population growth rates is driven primarily by variation in survival rates for species with slower life histories such as mammals (Oli & Dobson, 2003) and birds (Saether & Bakke, 2000), we parameterised the simulated population dynamics of 21 long-lived

Ricker logistic
Linear (compensatory), discrete-time relationship between the per capita rate of exponential population change (r) and population density Doncaster (2008); Ricker (1958) Stationarity (stationary) Opposite of nonstationarity; a dynamical system where the mechanisms generating fluctuations in population size do not change with time Dennis and Taper (1994); Turchin (2003) Stochastic Property of models estimating the probability of various outcomes while allowing for uncertainty in one or more parameters. In stable (stationary) systems, stochasticity is due to environmental factors; in chaotic systems, variability is caused by both internal (components of population structure like density feedbacks) and environmental factors Sinclair and Pech (1996)

Time series
Estimates of population abundance monitored at semi-regular intervals (e.g., years), collectively known as "census" Knape and de Valpine (2012)

Vital rate
See demographic rate a Density feedback should replace density dependence because, while used synonymously, the former abates conceptual and terminological confusion (Herrando-Pérez et al., 2012b).

TA B L E 1 (Continued)
F I G U R E 1 Scheme of the main elements of how density feedback operates in population dynamics (see Table 1 for a full glossary of terms indicated in italicized boldface). (i) a component density feedback can operate on survival probability (shown here as compensation) where survival declines as population size increases. (ii) Another common component density feedback operates on fertility, where the number of offspring per female decreases with increasing population size. (iii) A time series of abundance estimates ("census data") for a population captures an ensemble density feedback on the per capita rate of exponential population change (r) resulting from all component density feedbacks. In systems demonstrating stationarity, the underlying mechanisms (e.g., carrying capacity K) driving change in population size do not themselves shift over time. (iv) Plotting the rate of population change (r = N t + 1 /N t ) against population size (N t ) provides a way to measure the evidence for, and strength of, ensemble density feedback. In this representation, a Ricker logistic model estimates the linear slope between r and N t (a negative slope here indicates compensation, but a positive slope would indicate depensation). Where r = 0 intersects the linear Ricker logistic fit, the long-term mean carrying capacity (K) can be estimated if not trending upward or downward. The black arrows indicate that, under compensatory dynamics, a population tends to grow towards K when r > 0 (i.e., low N) and to decline from K when r < 0 (i.e., high N). (v) In this example, the system is in a state of nonstationarity because the K is declining over time.  Table 2). These species differ in their resilience to environmental change, and represent the slow end of the slow-fast continuum of life histories (Herrando-Pérez et al., 2012c). Here, high survival rates make it possible that reproductive efforts are dispersed over the lifetime of individuals (Gaillard et al., 1989). We chose this suite of species to cover a range of demographic types-it is the relative structure of the population and the particulars of the life histories that matter here, not the specifics of species A or B, or whether they are extant or extinct or live(d) in Australia or elsewhere. A full justification of the selection of our test species can be found in Bradshaw et al. (2021).

| Base (age-structured) population model
We developed the base population model for each test species as a stochastic (i.e., parameters resampled within their uncertainty bounds) Leslie transition matrix (M) following a prebreeding design (Caswell, 2001; Table 1). The Leslie transition matrix M has ω + 1 (i) × ω + 1 (j) elements (ages from 0 to ω years) for females only, where ω = maximum longevity. Fertility (m x ) occupied the first row of the matrix, survival probabilities (S x ) occupied the subdiagonal, and the final diagonal transition probability (M i,j ) was S ω for all species-except Vombatus ursinus (VU; common wombat), Thylacinus cynocephalus (TC; thylacine), and Sarcophilus harrisii (SH; devil), for which we set S ω = 0 to limit unrealistically high proportions of old individuals in the population given the evidence for catastrophic mortality at ω for the latter two species (Cockburn, 1997;Holz & Little, 1995;Oakwood et al., 2001). Multiplying M by a population vector n estimates total population size (Σn) at each forecasted time step (Caswell, 2001). We parameterised the base model with n 0 = ADMw for a closed population (dispersal = 0), where w is the right eigenvector of M (stable stage distribution), and A is the surface area of the study zone (A = 250,000 km 2 ), so that the species with the lowest n 0 would have an initial population of at least several thousand individuals at the start of the simulations. Based on theoretical equilibrium densities (D, km −2 ) calculated for each taxon (Bradshaw et al., 2021), we set the species-specific carrying capacity K = DA.
We ran projections of the base model to 40 generations (40⌊ G⌉; see Section 2.6) per simulated population such that:

TA B L E 2
Taxonomy and life-history characteristics of the 21 test species (all native to Australia) used to simulate age-structured populations and time series of population abundance (Bradshaw et al., 2021). where (v T M) 1 is the dominant eigenvalue of the reproductive matrix R derived from M, and v is the left eigenvector of M (Caswell, 2001).

| Setting the survival modifier
We simulated a component compensatory density-feedback function by forcing a reduction modifier (S red ) of the age-specific survival (S x ) vector in M according to total population size Σn: where the a, b, and c constants for each species are adjusted to produce a stable population on average over 40 generations (40⌊ G⌉; see above) Traill et al., 2010). This formulation avoids exponentially increasing populations and optimizes transition matrices to produce parameter values as close as possible to the maximum potential rates of population increase (r m ), therefore ensuring that long-term population dynamics are approximately stable at the species-specific carrying capacity.
The total projection length in years (q)

| Varying uncertainty in survival
In each projection and annual time step, the survival vector S x was resampled following a β distribution assuming a 5% standard deviation of each S x and a Gaussian-resampled fertility vector m x . We tested that increasing the standard deviation on juvenile survival (Barraquand & Yoccoz, 2013;Hilde et al., 2020) had no effect on our conclusions (see Appendix 2 and Section 3).

| Catastrophe function
For each species, we added a catastrophic (density-independent) mortality function to the transition matrix M and scaled it to generation length among vertebrates (Reed et al., 2003): where p C = probability of catastrophe was set at 0.14 given this is the mean probability per generation observed across vertebrates (Reed et al., 2003). Once invoked at probability C, a binomial β-resampled proportion centred on 0.5 to the β-resampled survival vector induces a ~ 50% mortality event for that year (Bradshaw et al., 2013). A catastrophic event is defined as "… any 1-yr peak-to-trough decline in estimated numbers of 50% or greater" (Reed et al., 2003). The catastrophe function essentially recreates the demographic effects of a densityindependent process such as extreme weather events, fires, or disease outbreaks.

| Adding a component feedback in fertility
We deliberately avoided applying density-feedback functions to fertility to isolate the component feedback to a single demographic rate (survival, see above). However, we also tested whether splitting the compensatory feedback between survival and fertility altered our results and conclusions (see Appendix 3 for justification and test outcomes). Our conclusions remained the same without or with a density feedback on fertility (Section 3).

| Generating nonstationarity
Nonstationarity is defined as a property of a long-term population trend whereby the density-dependent and -independent mechanisms generating fluctuations in population density themselves change through time (Turchin, 2003;  proportional offtake in the abundance vector (n) such that the population declined on average over the projection interval (two rates of population decline considered); (iv) variable but declining carrying capacity; (v) catastrophe survival function increased to produce a stable long-term population trend (r ≅ 0) over 40 generations with a null density feedback on survival. These nonstationary mechanisms recreate real situations experienced by wild populations of largebodied carnivores and herbivores exposed to temporal changes in food resources or mortality events resulting from disease outbreaks or harvesting.

| Measuring nonstationarity in abundance time series
To ascertain the degree of nonstationary in each simulated abundance time series (Section 2.6) across all demographic scenarios (Section 2.7), we calculated the mean and variance of return time (T R )-defined as the time required to return to equilibrium following a disturbance (Berryman, 1999). We calculated the mean and variance of return time for each generated abundance time series as: (2) where T R is the mean T R across M steps of the time series. For each where S C m is the number of complete time steps taken before reaching where N is the abundance mean across all time steps in the time series (a proxy for carrying capacity), N p is the population size prior to crossing N, and N a is the population size after crossing N. The variance of T R is: ries is considered to be highly nonstationary (Berryman, 1999).
See Appendix 1 and Figures S1-S3 for how these the perturbations imposed in the demographic scenarios altered indices of nonstationarity.

| Simulating time series of population abundance
From the base model M that incorporates age structure, density feedbacks on survival, catastrophic events, and varying carrying capacity as described above, we generated multiannual abundance time series up to 40 generations for each species (Section 2.2; Equation 1). We standardized projection length to 40 generations because there is strong evidence that the length of a time series (q) dictates the statistical power to detect an ensemble densityfeedback signal in logistic growth curves Knape & de Valpine, 2012). Here, we summed the n abundance vector over all age classes to produce a total population size N t,i for each year t of each projection i. We rejected the first generation of each projection as a burn-in to allow the initial (deterministic) age distribution to calibrate to the stochastic expression of stability under compensatory density feedback.

| Demographic scenarios
We generated 10,000 abundance time series over 40 generations (Sections 2.2 and 2.6) for each of the 21 test species ( Below, we present the nine demographic scenarios (summarized in Table 3), and then we describe the measurement of ensemble and compensatory feedbacks (statistical support in Section 2.8 and strength in Section 2.9) from each simulated time series across scenarios. Our set of scenarios emulate true nonstationary processes (Section 2.4; Appendix 1) often shaping the long-term population dynamics of large mammals through density-independent (catastrophic and harvest) mortality and variation in carrying capacity.
Our focus is on whether those processes erode the density-feedback

| Stochasticity in demographic rates (Scenario i)
Scenario i: Population subjected to the stochasticity imposed by resampling demographic rates in the Leslie matrices (Section 2.2) (Dennis et al., 2006). This is the only scenario where we impose no catastrophic mortality events.

| Catastrophic mortality (scenarios ii and iii)
Scenario ii: As in Scenario i, but with generationally scaled catastrophes centered on 50% mortality, leading to population stability (r ≅ 0). Compared to Scenario i, Scenario ii tests the hypothesis that density-independent catastrophes imposing process error erode the density-feedback signal from time series of abundance (Abadi et al., 2012;Knape & de Valpine, 2012).
Scenario iii: As in Scenario ii, but with an additional, single "pulse" perturbation of 90% mortality applied across all ages at 20 generations to alter the population age structure-this tests the hypothesis that large "resets" of population size modify the underlying component dynamics so abruptly via highly modified age structure that the ensemble signal is eroded (Hoy et al., 2020;Turchin, 2003 ;Wu et al., 2007).

| Harvest-like mortality (scenarios iv and v)
Scenario iv: A "harvest"-like scenario where a consistent proportion of individuals is removed from the n abundance vector at each time step to produce a weakly declining population on average (r ≅ −0.001) (Bargmann et al., 2020;Bergman et al., 2015) (this scenario also includes the castastrophic mortality function described in Scenario ii).
Scenario v: As in Scenario iv, but with a strongly declining popu-

| Absence of component density feedback on survival (Scenario ix)
Scenario ix: This is the only scenario where we imposed no component density feedback on survival, testing the hypothesis that in populations exposed to high density-independent process error, false detection of an ensemble signal can occur even when component feedback is weak or absent (Knape, 2008 produce a stable population on average (r ≅ 0) over 40 generations, and removed the component density-feedback on survival by setting the survival reduction parameter S red to 1 in all iterations.

| Measuring ensemble density feedbacks
For each simulated time series, we applied four phenomenological models to quantify both the statistical evidence of the ensemble compensatory density feedback and the strength of such a feedback as follows:

| Phenomenological models
The phenomenological models included four variants of the general logistic growth curve (Verhulst, 1838) following :  (Ricker, 1954), and (4) Gompertz-logistic (Nelder, 1961), where N t on the right side of Equation 8 is replaced with log e (N t ). The latter two models represent alternative situations where population growth rate varies in response to unit (Ricker) or order-of-magnitude (Gompertz) changes in population density (Herrando-Pérez et al., 2012b).

| Strength of ensemble density feedback
We estimated the strength of the ensemble density-feedback as the negative of the slope ̂ estimated from the Gompertz-logistic model (under compensation, ̂ will always be < 0, so the lower the ̂ , the stronger the compensatory feedback). We used the Gompertzlogistic ̂ , instead of the Ricker-logistic ̂ , to estimate this strength because only the former characterizes the multiplicative nature of demographic rates (Doncaster, 2008;Herrando-Pérez et al., 2012a).

| Statistical evidence for ensemble density feedback
We calculated the relative likelihood of the four phenomenological models fitted to each time series by means of the Akaike's information criterion (AIC) corrected for finite number of samples (AIC c ) (Sugiura, 1978) in a multimodel inferential framework (Burnham & Anderson, 2002). Across the four models, we ranked the statistical evidence for an ensemble density-feedback Pr(density feedback) as the sum of AIC c weights (wAIC c = model probability) for the Ricker-and Gompertz-logistic models (i.e., ΣwAIC c -density feedback), and the evidence for a lack of such feedback as the sum of AIC c weights for random walk and exponential growth (i.e., ΣwAIC cdensity independence)-where ΣwAIC c -density feedback + ΣwAIC cdensity independence = 1 (Burnham & Anderson, 2002). This follows the logic that the more the slope between the per-capita rate of change (r) and abundance (N t ) (Ricker model)

| Correlating ensemble versus component density feedbacks
We

| Magnitude of ensemble density feedbacks
Bootstrapping across all species, the reduction in ensemble densityfeedback strength measured as Gompertzβ was greatest in Scenarios iv and v where we imposed population declines of r ≅ −0.001 and Strength of ensemble compensatory density feedback across demographic scenarios. Bootstrapped (10,000 uniform resamples between 95% confidence limits) across 21 test species (detailed in Table 2) of the strength of ensemble compensatory density feedback (Gompertz-β) among scenarios (detailed in Table 3). Midpoints indicate means, and error bars are the interquartile ranges. Demographic scenarios include carrying capacity K fixed (K fixed ; Scenario ii), a pulse disturbance of 90% mortality at 20 generations (20G; Scenario iii), weakly declining (r ≅ −0.001; Scenario iv) and strongly declining (r ≅ −0.01; Scenario v) populations, K varying stochastically (K stoch ) around a constant mean with a constant variance (Scenario vi), K varying stochastically with a constant mean and increasing variance (K stoch ↑Var; Scenario vii), and K varying stochastically with a declining mean and a constant variance (↓K stoch ; Scenario viii).

F I G U R E 3
Decoupling of ensemble and component density feedbacks in demographic scenarios with and without catastrophic mortality. Relationship between strength of ensemble (slope coefficient β of the Gompertz-logistic model × [−1] in the time series) and component (1 -the modifier S red on survival in the Leslie transition matrix) density feedback for: Scenario i (pink; stochastic mortality, no catastrophic mortality, stable K) and Scenario ii (grey: stochastic mortality, catastrophic mortality, stable K). Fitted curves across species are exponential plateau models of the form y = y max − (y max − y 0 )e −kx . Shaded regions represent the 95% prediction intervals for each scenario. Each scenario includes 21,000 simulated time series of abundance (10,000 for each of 21 species; Table 2). Also shown are the mean probabilities of median density feedback (Pr(density feedback): sum of the Akaike's information criterion weights for the Ricker-and Gompertz-logistic models across time series (ΣwAIC c -density feedback) relative to the weights of two density-independent models (random and exponential).
r ≅ −0.01, respectively, relative to the baseline Scenario ii (r ≅ 0) with population stability over time (Figure 2). The next largest reductions in the ensemble signal occurred in Scenarios iii (pulse perturbation at 20 generations) and viii (stochastically varying carrying capacity declining over time) (Figure 2). Lastly, Scenarios vi (stochastically varying carrying capacity around a long-term stable average) and viii (stochastically varying carrying capacity around a long-term stable average, with increasing variance over time) had similar ensemble feedback strengths relative to the base Scenario ii (Figure 2). Clearly, only harvest-like mortality (Scenarios iv and v) dampens the strength of compensatory density feedbacks on population growth rates. ( Figure 4). Noticeably, some abundance time series experienced F I G U R E 4 Decoupling of ensemble and component density feedbacks in demographic scenarios with catastrophic mortality and with catastrophic mortality + pulsed mortality and harvesting (see Figure 6). Relationship between strength of ensemble (slope coefficient β of the Gompertz-logistic model × [−1]) and component (1 -the modifier S red on survival) density feedback for: Scenario iii (green: pulse disturbance of 90% mortality at 20 generations); Scenario iv (red: weakly declining population at r ≅ −0.001); and Scenario iv (blue: strongly declining population at r ≅ −0.01). Each scenario includes 21,000 simulated time series of abundance (10,000 for each of 21 species; Table 2). Fitted curves across species are exponential plateau models of the form y = y max − (y max − y 0 )e −kx . Shaded regions represent the 95% prediction intervals for each scenario. Also shown are the mean probabilities of median density feedback (Pr(density feedback): sum of the Akaike's information criterion weights for the Ricker-and Gompertz-logistic models across time series (ΣwAIC c -density feedback) relative to the weights of two density-independent models (random and exponential). depensation or "Allee effects" (population growth rate increasing with population size; Table 1). For these two harvest-like scenarios (Scenarios iv and v), the 95% confidence interval of the en-  (Figure 8 and Figure S8). The former correlations indirectly reinforce the observation that density-independent mortality is a stronger driver of component-ensemble density-feedback decoupling than fluctuating resources (Subsection 3.2.1) as the variation in the magnitude of density feedbacks is more responsive to variation in carrying capacity than to density-independent mortality.  Table 2). Fitted curves across species are exponential plateau models of the form y = y max − (y max − y 0 )e −kx . Shaded regions represent the 95% prediction intervals for each scenario. Also shown are the mean probabilities of median density feedback (Pr(density feedback): sum of the Akaike's information criterion weights for the Ricker-and Gompertz-logistic models across time series (ΣwAIC c -density feedback) relative to the weights of two density-independent models (random and exponential).

| DISCUSS ION
Our simulations reveal several new insights into how densityfeedback signals in population growth rates and those operating on vital rates can be decoupled. First, we discovered that the estimated strength of density feedbacks from abundance time series are particularly sensitive to density-independent mortality that produces long-term declines in population size. In other words, logistic models are unlikely to reveal density feedback in harvested populations that are declining, even when strong component feedbacks exist.
Therefore, attempting to measure density feedbacks in such populations only from time series of abundance would be unlikely to bear fruit. On the contrary, estimated feedback strength is much less sensitive to moderate fluctuations in carrying capacity.
Second, the statistical detection of density feedbacks in abundance time series is robust in the face of even pronounced nonstationarity. It is essential here to distinguish the detection from the strength of the feedback itself-the former is based on the statistical evidence that phenomenological models provide more support for a relationship between rate of change and population density than not , whereas the latter indicates the magnitude of the slope of that relationship (Herrando-Pérez et al., 2012c).
Third, the concern that density-independent processes can invoke false evidence of ensemble signals of compensation are not borne out by our simulations, at least with respect to density-independent mortality not leading to declining population size. Our results therefore lend credence to the application of phenomenological (logisticgrowth) models to studies addressing the long-term effect of vital rates on population abundance, provided there is enough information available (i.e., population censuses over long periods) for describing population trends.
The relative magnitude of density-dependent and -independent mechanisms and their characterization and detection with logistic models will vary from population to population. For instance, variation in survival probability can be entirely driven by variation in climatic conditions and density-independent predation (Hebblewhite et al., 2018). In one of the best-studied systems in this regard, Soay  (Lieury et al., 2015) and spatial heterogeneity in population growth rates (Thorson et al., 2015).
Indeed, examining the nuances of spatial heterogeneity and the exchange of individuals among populations would require a completely different modeling framework than the one we constructed  Figure 2) and Scenario ix (grey: without compensatory density feedback). Each scenario includes 21,000 simulated time series of abundance (10,000 for each of 21 test species, Table 2). Probabilities of density feedback (Pr(density feedback) = sum of the Akaike's information criterion weights for the Ricker and Gompertz models relative to the weights of two density-independent models (random and exponential)) calculated across simulations gave median Pr(density feedback) = 0.994 and 0.322 for the two stable scenarios with (Scenario ii) and without (Scenario ix) component feedback on survival, respectively.
here. Other disrupting phenomena such as fluctuating age structure (Hoy et al., 2020), environmental state shifts (Turchin, 2003;Wu et al., 2007), and sampling error (Knape & de Valpine, 2012) were implicit in our modeling framework. In addition, by standardizing the spatial extent and population densities at the beginning of all projections, and by including known sampling and process errors, our models quantify the contributions of nonstationarity and other forms of density-independent change to vital rates.
Another caveat is that simulating closed populations might have potentially inflated our capacity to detect the component signal in abundance time series, because permanent dispersal could alleviate per-capita reductions in fitness as a population nears carrying capacity. We also limited our projections to a standardized 40 generations across species, but even expanding these to 120 generations resulted in little change in the stationarity metric ( Figure S9). Complementary studies focusing on the faster end of the life-history continuum could provide further insights, even though our range of test species still produced a life-history signal of the strength and stationarity of component ( Figure S10) and ensemble density feedbacks (Figures S11 and S12) that declined with increasing generation length. However, this relationship faded when the trajectories simulated declines through proportional removal of individuals. Indeed, both evidence for (Holyoak & Baillie, 1996), and strength of (Herrando-Pérez et al., 2012c), ensemble density feedback generally increase along the continuum of slow to fast life histories, because species with slow life histories are assumed to be more demographically stable when density compensation is operating (Saether et al., 2002).

F I G U R E 7
Strength of ensemble density feedback in demographic scenarios with catastrophic mortality, catastrophic mortality with pulsed mortality, and two types of harvesting. Relationship between strength of ensemble density feedback (slope coefficient β × [−1] of the Gompertz-logistic model) and the stationarity index T R ∕ Var T R across 21 test species over 40 generations for four demographic scenarios: (a) Scenario ii: carrying capacity (K) fixed, (b) Scenario iii: a pulse disturbance of 90% mortality at 20 generations, (c) Scenario iv: weakly declining population at r ≅ −0.001, and (d) Scenario v: strongly declining population at r ≅ −0.01. Each scenario includes 21,000 simulated time series of abundance (10,000 for each of 21 species, Table 2). Fitted curves across species exponential plateau models of the form y = y max − (y max − y 0 )e −kx . Shaded regions represent the 95% prediction intervals for each type. ρ med are the median Spearman's ρ correlation coefficients for the relationship between the ensemble strength and stationarity index across species (resampled 10,000 times; see Figure S8 for full uncertainty range of ρ in each scenario).

ACK N OWLED G M ENTS
We F I G U R E 8 Strength of ensemble density feedback in demographic scenarios with catastrophic mortality and fixed carrying capacity versus three types of fluctuating carrying capacity and no catastrophic mortality. Relationship between strength of ensemble density feedback (slope coefficient β × [−1] of the Gompertz-logistic model) and the stationarity index T R ∕ Var T R across 21 test species over 40 generations for four demographic scenarios: (a) Scenario ii: carrying capacity (K) fixed, (b) Scenario vi: K varying stochastically (K stoch ) around a constant mean with a constant variance, (c) Scenario vii: K varying stochastically with a constant mean and increasing variance (K stoch ↑Var), and (d) Scenario viii: K varying stochastically with a declining mean and a constant variance (↓K stoch ). Each scenario includes 21,000 simulated time series of abundance (10,000 for each of 21 species. Fitted curves across species exponential plateau models of the form y = y max − (y max − y 0 )e −kx . Shaded regions represent the 95% prediction intervals for each type. ρ med are the median Spearman's ρ correlation coefficients for the relationship between the ensemble strength and stationarity index across species (resampled 10,000 times; see Figure S8 for full uncertainty range under each scenario).

O PEN R E S E A RCH BA D G E S
This article has earned Open Data and Open Materials badges. Data and materials are available at [https://github.com/cjabr adsha w/ Densi tyFee dback Sims].

DATA AVA I L A B I L I T Y S TAT E M E N T
All data files and R code are openly available at https://github.com/ cjabr adsha w/Densi tyFee dback Sims.